Objective Similarity and Mental Representation
Alistair M. C. Isaac
June 6, 2012
Abstract
The claim that similarity plays a role in representation has been philosophically discredited. Psychologists, however, routinely analyze the success of mental representations for guiding behavior in
terms of a similarity between representation and the world. I provide a foundation for this practice by developing a philosophically responsible account of the relationship between similarity and
representation in natural systems. I analyze similarity in terms of the existence of a suitable homomorphism between two structures. The key insight is that by restricting attention to only those
homomorphisms induced by causal processes, we can solve two philosophical problems with a single
assumption. First, causal structure provides an adequate source for the bias required to ensure the
similarity relation is non-trivial; second, it provides an adequate source for the directionality required
to move from similarity to representation. I defend this account against objections by Goodman
and van Fraassen and demonstrate that it is indeed the account of similarity’s role in representation
assumed by psychological practice.
1
Introduction
How do humans reason successfully about the world? A standard answer is that we represent the world
and reason about it by manipulating these mental representations. But what special feature of a mental
representation allows us to use it to reason about the world? It can’t be the fact that it is a representation
simpliciter. I can stipulate that this pencil represents Canada, but no amount of manipulation of the
pencil will tell me the size of the population of Canada. So, what special feature makes a mental
representation different from the pencil?
The classical answer to this question is that the mental representation is similar to the object it
represents. Aristotle used the analogy of the signet ring to explain how this works. When a signet ring
is pressed into warm wax, the wax receives the form of the ring, but not the matter. Aristotle argued
that perceptual organs behave just like the wax. When the eye sees an object, it takes on its form, but
not its matter. Since the mental representation has a similar form to the external object, it can be used
to reason about that object (De Anima, Bk. II, Ch. 12).
From the perspective of philosophy, the similarity analysis of mental representation has fallen on hard
times. Already in the seventeenth century, the idea that the properties of the mental and those of the
physical are comparable was completely discredited. By the twentieth century, the idea that similarity
could explain representation in any context was also undermined. Nelson Goodman argued forcefully
against the value of similarity as a philosophical concept, exposing it as “a pretender, an imposter, a
quack” (Goodman, 1970, p. 437).
1
Unfortunately, the psychologists don’t seem to be listening. Implicit appeals to the structural similarity between a mental representation and the external object it represents appear repeatedly in research
on perception and mental imagery (for summaries of some experimental findings, see e.g. Shepard, 1975,
or Kosslyn et al., 2006). Even those few holdouts who reject the theoretical conclusions of the imagery
literature (that there are distinctively picture-like representations) find themselves forced to allow for
some kind of structural similarity between symbolic representation and object in order to account for the
data (e.g. Pylyshyn, 2002).
The purpose of this paper is to reconcile psychological practice with philosophical theory. In particular,
the task is to define a notion of objective similarity which can explain the informativeness of mental
representations while avoiding Goodman-style counter-arguments. The key step is to allow ourselves the
assumption of causal structure in the world. It turns out that if we assume the causes and categories of
the current scientific view of the world, a natural notion of similarity falls out of them (Section 2).
The lynchpin of the argument that similarity cannot underwrite representation is directionality. Similarity is symmetrical, while representation is directed. Here again the assumption of causal structure
can help. By combining the directionality of causal relations with objective similarity, we can produce
a definition of representation in nature, or N-representation. N-representation is just another relation
existing in nature, and does not depend in any way on human purposes (if it did, it would be explanatorily impotent for understanding mental representations). Van Fraassen has taken Goodman’s arguments
to imply that no such natural representational relation is possible, but we’ll see that his argument only
applies to a very restricted class of representations, not the concept in general (Section 3).
Finally, we’ll conclude by surveying psychological theories of mental representation. We’ll see that
several prominent views take mental representations to be instances of N-representation. These internal
N-representations graduate to full blown mental representations once they function to guide action. We’ll
conclude with a brief analysis of the source of misrepresentation on this account. The objective similarity
between mental representations and objects in the world helps us to reason about the world, but it still
leaves room for us to make mistakes (Section 4).
2
Objective Similarity
There are two natural strategies for defining similarity: property-based and isomorphism-based.1 The
property-based strategy takes two objects to be similar when they share properties. The isomorphismbased strategy takes two objects to be similar if a mapping between their component parts exists. The
account developed here is an isomorphism-based account as this conforms more closely to the use of
terms in the psychological literature. Nevertheless, it is worth stressing that the choice of properties or
isomorphisms is essentially one of convenience. A property-based account can be recast in isomorphism
terms, and an isomorphism account can be characterized in terms of shared second-order properties.
The fundamental challenge for any definition of objective similarity is accounting for bias without
appealing to a human agent. After a survey of this problem, we’ll examine some examples of similarity
in nature, before introducing the full definition of objective similarity.
1 A note on terminology. I will use the term “isomorphism” for the general notion of a structure preserving mapping
and “homomorphism” for the specific type of structure preserving mapping introduced in Section 2.3. Although somewhat
mathematically lax, this choice follows the use of psychologists, who frequently use the term “isomorphism” when context
makes clear that they mean “homomorphism” (as in the quotes from Gallistel and Shepard in Section 4.1).
2
2.1
The Challenge: Whence the Bias?
When applied naı̈vely, both property and isomorphism-based analyses of similarity suffer from exactly
the same problem: too many similarities. In order to avoid this problem some kind of bias needs to
be introduced. Without any source of bias, everything is similar to everything else, and similarity is a
trivial notion. By privileging some properties (or mappings) over others, bias can reduce the number of
similarities, making the concept non-trivial.
For property-based accounts of similarity, the too many similarities problem is a consequence of the
Ugly Duckling Theorem. The Ugly Duckling Theorem says that if properties are extensionally defined
and closed under complementation and negation (i.e. form a Boolean lattice), then any two objects share
the exact same number of properties. This means that any two objects are exactly as similar as any other
two objects. To see this, consider a finite universe of N objects. There are 2N ways to group them into
sets, each of which corresponds to a property. Code each set with an N digit binary string where each
digit stands for the membership, 1, or not, 0, of the respective object in that set; there will be 2N such
strings. Pick two objects arbitrarily and order the strings lexicographically with respect to their digits.
On the first quarter of the strings, they will agree on membership, i.e. the strings will begin 11 . . ., on the
next two quarters they will disagree (strings 10 . . . and 01 . . .), but on the final quarter they will agree
again (strings beginning 00 . . .). So the two objects agree on exactly 1/2 of their properties, i.e. on 2N −1
properties. Since they were chosen arbitrarily, the result holds for any two objects.
The Ugly Duckling Theorem is near trivial, but it makes a crucial point. If you want to say two
objects are similar because they share properties, you need to privilege some properties over others. If
you accept all boolean combinations of properties as themselves properties of equal status, your claim
is trivial. This result was proved (and named) in Watanabe (1969), but had surely been known for a
long time before that. The first edition of Goodman (1976) alludes to it and Goodman (1970) gives the
argument explicitly, without any evidence of influence from Watanabe.
For isomorphism-based accounts of similarity, the Newman Problem plays the same role as the Ugly
Duckling Theorem. In response to Russell’s early structural realism, Newman (1928) pointed out that
an isomorphism exists between any two sets of the same cardinality. If we demand that the isomorphism
preserve relations found in one of the structures, it will always be mathematically possible to define
relations in the second structure which satisfy it, so long as the cardinality of the two structures is the
same. McLendon (1955) pointed out that the problem is even worse. Since any set of objects can be
subdivided into arbitrarily many subsets, an isomorphism exists between structures defined over any two
sets regardless of cardinality. Since some such isomorphism exists between any two structures, the naı̈ve
claim that two structures are similar because an isomorphism exists between them is trivially true of any
two structures.
Again, the moral is that some source of bias is needed. We must privilege some isomorphisms over
others, or some relations over others, in order for the claim that similarity occurs whenever two objects
are isomorphic to be non-trivial. In the account of similarity developed here, the properties and relations
of the scientific view of the world provide the source of our bias. In order to see how this will work, let’s
first look at some examples.
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2.2
Examples of Similarity in Nature
Aristotle’s signet ring provides a good starting point for thinking about similarity in nature. Of course,
the ring wearer intends the wax to take the shape of the ring, but that it does so does not depend on his
wishes, but on the properties of the ring and the wax. Examples of similarity of shape such as this can
be found everywhere in nature. For instance, the relationship between each of these two objects might
plausibly be considered one of similarity:
1. a footprint and the foot which made it
2. a shadow of a tree and the tree which cast it
3. a fossil of a mollusc and the original mollusc
In each case, there are two shapes, and some gross similarity between the shapes in virtue of the fact that
one caused the other. In the case of the footprint and the fossil, the causal relationship involves direct
contact, or at least spatial coincidence. In the case of the shadow, the fact that light travels in straight
lines ensures that rays from a single source will induce a projection of any three dimensional object onto
the surface behind it.
Of course, direct contact is not a necessary condition for sameness of shape. Two shapes may be
similar in virtue of each having separate contact with the same object, for instance
4. two footprints, both made by the same foot
More complex causal relations can also induce similarities of shape. Consider for example
5. the pattern of colors on a butterfly wing and that on the wing of its offspring
In this case, there is a similarity of shape (and color), but the causal process involves the transmission of genetic material, epigenetics, sameness of environment and diet, and many additional factors.
Furthermore, it is not just shape which is similar in this case, but also behavior and life cycle.
In each of these examples, we might argue that the two similar objects share the same properties,
and that their similarity is reflected in the fact that these properties take the same value for each object.
For instance, the foot is 11 inches long, and its footprint is also 11 inches long; or, the butterfly’s wing is
yellow and its offspring’s wing is also yellow. But similarity may also hold purely at the structural level;
consider the relationship between
6. the (relative) width of a tree ring and the (relative) amount of rain during the year in which it was
formed
7. fluctuations in the ratio of black to white peppered moths and fluctuations in the ratio of dark to
light colored tree bark in Sherwood Forest
In 6, the properties at issue are different: widths and volumes are different sorts of properties. The
similarity relation holds not between the sets of properties, but between the pattern of instances of one
property and the pattern of instances of another. This is the more general type of similarity which can
be found in nature, and instances in which the properties happen to be the same are just special cases of
it.
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2.3
Objective Similarity Defined
Nature is filled with structures. An animal, a population, a spatio-temporal region: each of these has
internal structure, i.e. a pattern of parts with properties standing in various relations to each other.
The same natural structure may be characterized in many different ways—in the present project, we
allow ourselves any method of doing so which can be found in the scientific description of the world, and
deny any which is artificially or arbitrarily constructed. Science then becomes the source of bias which
privileges some relations and mappings over others, avoiding the Newman Problem.
Formally, we can represent structures as set theoretical objects, e.g. hX , R1 , R2 , R3 , . . .i, where X
is the set of parts and R1 , R2 , R3 , . . . are n-ary relations which hold between them (a property is just a
unary relation). We say a structure is trivial if all R1 , R2 , R3 , . . . are either empty or equivalence relations
over all X . The natural similarity relationship between two set theoretical objects is a homomorphism,
or structure preserving mapping. If hX , Ri and hY, Si are two structures with R and S binary relations,
then a map φ : X −→ Y is a homomorphism just in case xRx0 if and only if φ(x)Sφ(x0 ) for all x, x0 ∈ X .
Of particular interest are structural relationships defined by orderings. Consider, for example, the
case of tree rings and rain. Take hR, ≤C , ≤W i to be the set of tree rings with two orderings defined on
them. For r, r0 ∈ R, r ≤C r0 iff r0 is at least as far from the center of the tree as r, and r ≤W r0 iff r0 is
at least as wide as r. Take hW, T , V i to be the collection of annual rainfalls in the forest during the
life of the tree. For w, w0 ∈ W, w T w0 iff w0 occurred at least as late as w and w V w0 iff w0 was at
least as great in volume as w. There exists a map ψ : W −→ R such that w T w0 iff ψ(w) ≤C ψ(w0 )
and w V w0 iff ψ(w) ≤W ψ(w0 ). Furthermore, it is in virtue of the inverse of this map, ψ −1 , that
dendrochronologists are able to learn about past rainfall by examining the rings of old trees.
The tree rings and rainfall, just like a foot and its footprint, are similar as the result of a direct causal
relationship. We might plausibly claim that the quantity of rainfall in a given year caused the width
of the respective ring, or the shape of the foot caused the shape of the footprint. Let us say that a
homomorphism which exists as a consequence of a causal process is induced by that process.
Homomorphisms which are induced by causal processes establish similarity relations, but they do not
exhaust the similarities which can be found in nature. Consider two examples. First, the relationship
between a foot and a plaster cast made from one of its footprints. The shape of the footprint is induced
by its causal relationship to the foot, while the shape of the cast is induced by its causal relationship
to the footprint. It doesn’t matter whether these two causal relations are in any way related (in this
case, plausibly, they are not)—what matters for ensuring similarity is that the structure preserved by the
second causal process is the same as the structure preserved by the first causal process.
Second, consider the relationship between two footprints made by the same foot. In this case, there
exists a structure preserving homomorphism between the two footprints, but it is not induced directly
by a causal process. Rather, it is the fact that both footprints stand in the same causal relationship to a
third structure, the foot, which ensures they are similar.
These considerations motivate the definition of a similarity map.
Similarity Map – call a homomorphism φ from structure A into structure B a similarity
map if any one of the following hold
• φ was induced by a causal process in which the structure of A causally influenced the
structure of B.
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• A = B.
• There exist maps ψ and ψ 0 and structure C such that ψ is a similarity map from A into
C, ψ 0 is a similarity map from C into B, and φ = ψ 0 ◦ ψ.
• There exist maps ψ and ψ 0 and structure C such that ψ is a similarity map from C into
A and ψ 0 is a similarity map from C into B such that ψ 0 = φ ◦ ψ.
The first condition establishes the base case. The second condition ensures reflexivity. The third condition
is motivated by the example of the cast and the foot. Since the definition is inductive in structure, it
allows for, e.g. footprints made by the plaster cast to be similar to the original foot, or the pattern on a
butterfly’s wing to be similar to that on the wing of its great-great-grandmother. The fourth condition is
motivated by examples like footprints made by the same foot, or widths of tree rings in the same forest.
Is the set of similarity maps symmetric, i.e. if θ is a similarity map from B into A, is θ−1 a similarity
map? Symmetry follows from clauses 2 and 4. To see this, consider the instance of clause 4 for which
C = B. By stipulation the similarity map ψ (= θ) exists. ψ 0 exists and is a similarity map by clause 2
and the stipulation that C = B. Then φ (= θ−1 ) is a similarity map as it satisfies the constraint that
ψ 0 = φ ◦ ψ since ψ 0 = θ−1 ◦ θ.
Finally, notice that the similarity which holds between two structures in nature is established by a
partial map between them. Rings are not all there is to a tree; shape of the sole is not all there is to a
foot; a shadow does not preserve the full three dimensional structure of the object which cast it. These
considerations motivate the definition of objective similarity:
Objective Similarity – Two natural structures A and B are objectively similar if there exist
non-trivial substructures A0 ⊆ A and B 0 ⊆ B such that
1. There exists a homomorphism φ : A0 −→ B 0
2. The homomorphism φ is a similarity map.
Note that this definition applies to all the above examples of similarity in nature. For example, the relevant
structure preserved in cases of similar shape is relative distance between points, the causal process is direct
contact (the footprint), obstruction (the shadow), or genetic transmission (the butterfly).
The insistence that the substructures be non-trivial rules out trivial or vacuous similarities where
there is a causal relationship but no structure preserving map. A cocoon is not similar to a moth, a hole
in the ground is not similar to a stick of dynamite, and a mosquito bite is not similar to a mosquito. In
each case, there is a causal relationship, but no shared structural relations between cause and effect, so
only trivial homomorphisms exist between the two structures.
By allowing ourselves to assume that there are natural (i.e. scientific) categories and causal relations
in the world, we’ve defined a notion of similarity which is objective, in the sense that it does not depend
upon the judgments of any particular agent. We’ll see in Section 4 that this is the exact notion of
similarity employed by many psychologists in analyzing mental representations. Before we get there,
however, we need to discuss the general relationship between objective similarity and representation in
nature.
6
3
From Similarity to Representation
The attempt to define representation in terms of similarity faces a fundamental problem: similarity
is symmetric, but representation is not. After discussing the classic presentation of this problem in
Goodman, we’ll define a type of representation which avoids it, N-representation, or representation in
nature. N-representation avoids Goodman’s worry by making use of the asymmetry in the causal relations
we’ve already allowed ourselves. Happily, then, we can develop a notion of representation from objective
similarity which requires no additional assumptions.
But is N-representation really representation? Van Fraassen thinks not. He argues that the considerations raised by Goodman demonstrate that not only directionality, but human purpose is required for
representation. I conclude the section by arguing that van Fraassen begs the question. By directing attention to examples from science rather than art, I demonstrate that N-representation shares qualitative
features with intuitive examples of representation.
3.1
The Challenge: Directionality
Nelson Goodman’s Languages of Art begins with a sustained argument that resemblance is inadequate
as a basis for representation. His fundamental point is that representation and symmetry have different
logical properties.
An object resembles itself to the maximum degree but rarely represents itself; resemblance,
unlike representation, is reflexive. Again, unlike representation, resemblance is symmetric: B
is as much like A as A is like B, but while a painting may represent the Duke of Wellington,
the Duke doesn’t represent the painting. Furthermore, in many cases neither one of a pair
of very like objects represents the other: none of the automobiles off an assembly line is a
picture of any of the rest; and a man is not normally a representation of another man, even his
twin brother. Plainly resemblance in any degree is no sufficient condition for representation.
(Goodman, 1976, p. 4)
Goodman argues that a) similarity is reflexive, but representation is not; b) similarity is symmetric, but
representation is not; and c) obvious examples of similarity are not instances of representation.
The properties of similarity Goodman identifies hold for objective similarity. Objective similarity is
reflexive due to the second clause of the definition of similarity map. The symmetry of objective similarity
follows from clauses 2 and 4, as discussed in Section 2.3. The examples of twin brothers and cars from
the same assembly line both satisfy the definition of objective similarity. But Goodman’s critique applies
to a bare notion of similarity, one which simply has the logical properties he identifies. Our notion of
objective similarity is much richer, with a built in source of directionality waiting to be exploited.
3.2
N-Representation and Its Properties
Causality is directional. In our definition of similarity map, we explicitly introduced reflexivity and
symmetry in order to derive a relation with the properties of similarity from causality. In order to
find those structures which are both similar and such that one represents the other, we simply need to
reintroduce that directionality. The fundamental idea here is that one structure can represent another
only if it is causally downstream from it.
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Causally Downstream – Given two natural structures A and B, A is causally downstream
from B whenever A 6= B and there exist structures C1 , C2 , . . . , Cn such that B causally
influenced C1 , Cn causally influenced A, and, for each 1 ≤ x ≤ n − 1, Cx casually influenced
Cx+1 .
Combining this directionality with the requirements of objective similarity we can define representation
in nature.
N-representation – Given two non-trivial natural structures A and B, A N-represents B
whenever both
1. A is objectively similar to B.
2. A is causally downstream from B.
Of course, we could have defined N-representation directly, without detouring through a general notion of
objective similarity. But in order to do so, we would have had to build the notion of structural similarity
into the definition nevertheless.
The first point to notice is that N-representation has the appropriate logical properties to count as a
representational relationship. It is neither reflexive nor symmetric; it is not satisfied by the examples of
similarity without representation provided by Goodman. For a more thorough argument that causally
induced homomorphisms satisfy the basic logical requirements for a theory of representation, see Bartels
(2006).
In what sense is N-representation “representation”? In the very intuitive sense of the re-presentation
of information. A relevant litmus test here is whether A can be used to learn about B. This was our
initial explanatory target—how can a mental representation be used to learn about the world? If A
re-presents information initially presented in B, then A can be used to learn about B. The insistence
on objective similarity ensures that A N-represents B implies that A contains information about B, i.e.
can be used to learn about B. The restriction to causal direction captures the asymmetrical nature of
information flow. In particular, if A represents B and B represents C, then B contains at least as much
information about C as A does. It is impossible to learn more about C from A than from B; increasing
causal distance from a structure means less information about that structure.
Consider, for example, the amount of information a forensic podiatrist can learn about a suspect’s
boot from a footprint. A footprint in damp, thick mud may leave lots of detail, the exact placing of
tread marks, irregularities in the wear of the sole, etc. Conversely, a footprint in very wet mud with low
viscosity may merely preserve the rough outline of the edge of the boot. The footprint in thick mud is
more similar to the boot than the footprint in wet mud precisely because it contains more information
about the boot. At a formal level, a larger substructure of the boot satisfies the objective similarity
relation with respect to the footprint in thick mud than with respect to the footprint in wet mud.
Now suppose the forensic podiatrist takes a plaster cast of the footprint. He may be able to learn
about the boot more easily from this cast than from the initial footprint (it’s easier to handle, to measure,
to transport; it won’t disappear with a heavy rain). But the cast can’t contain more information about
the boot than the initial footprint—the process of making the cast can’t “put in” any new information, it
can only extract information that’s already there. Consequently, the footprint itself will always be more
similar to the boot than the cast.
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This is just a specific instance of a general feature of information transfer familiar to anyone who has
copied a VHS tape. A copy of a VHS tape always looks worse than the original, more fuzziness and
noise, and a copy of a copy even more so. The same holds for photocopies: a copy of a document always
has more noise than the original document. Information degrades as it is transferred. This is a general
feature of the re-presentation of information captured by the notion of N-representation.2
3.3
Directionality Revisited
N-representation is a relation which exists in nature, independent of any human intentionality, agency, or
information use. It is a representational relation in the sense that a) it satisfies the logical constraints on
a such a relation (in particular, it is directed), and b) it captures the intuitive constraint that A represents
B implies that A can be used to learn about B. However, some have argued that mere directionality
and informativeness are not enough to constitute “representation.” In particular, van Fraassen explicitly
considers a relation very like N-representation, but denies that it constitutes a species of representation:
On the macroscopic level too we can think of processes that connect two situations separated
in time or space. These could be so correlated that it would be possible in principle to
get information about the one by inspecting the other—provided of course we knew of that
correlation! But that something could be done does not mean that it is done. That something
could be assigned a representational role does not mean that it has one. (van Fraassen, 2008,
p. 156)
For van Fraassen there is “no room for ‘representation in nature’, in the sense of ‘naturally produced’
representations that have nothing to do with conscious or cognitive activity or communication” (van
Fraassen, 2008, p. 24).
If van Fraassen is right, then N-representation is not representation. Of course, van Fraassen better
not be right, otherwise there would be no hope for a non-circular analysis of mental representation. If
conscious activity is required to explain any instance of representation, then mental structures can only
constitute representations if they are taken as such by a conscious agent. But the very concept of a
conscious agent already assumes representational mental structure. Regress. Where has van Fraassen
gone wrong?
Van Fraassen (2008) begins with an elaboration of the considerations of Goodman (1976). Whereas
we have focussed on the bare logical structure of representation as emphasized by Goodman, namely it’s
directionality, van Fraassen emphasizes the source of this directionality in Goodman’s examples, namely
human intent. For van Fraassen, representations are inherently perspectival, the source of this perspective
is the choice, goal, or intent of a particular representation user, and the mark of this perspectival quality
is the fact that representations are distorted with respect to their target.
For example, consider the ancient competition between Alcamenes and Phidias (reported in Gombrich,
1960, pp. 191–2). The two sculptors are competing to produce a statue of Minerva to be placed upon a
high pillar. Alcamenes produces a sculpture of a beautiful woman, while Phidias produces a sculpture
with stretched and enlarged features of the face. Yet, when the sculptures are placed upon the high
2 Pace Dretske’s unfortunately named “Xerox principle” (Dretske, 1981, p. 58), which denies this basic empirical feature
of information transfer.
9
pedestal, that of Phidias is much more pleasing to the eye when viewed from below. Distortion in the
sculpture made it a better representation when viewed from the relevant perspective.
Consider another example: a caricature of Richard Nixon. Such a caricature is more effective, succeeds
better at representing Nixon, when it emphasizes certain features, say the length of the nose, the beadiness
of the eyes, and the slackness of the jowls. Greater similarity with Nixon’s actual features does not equate
with better representation of Nixon in a political cartoon. Van Fraassen concludes from examples such
as these that distortion, more importantly the choice of which distortions to employ, is essential for
representation.
It seems then that distortion, infidelity, lack of resemblance in some respect, may in general
be crucial to the success of a representation. This does not rule out that resemblance in some
other respect may be required. Yet even when that is the case—and it may be a special
case—the choice of those respects in which resemblance or a specific kind of distortion is
required, and those for which just anything at all will do, will have to be seen as crucial as
well. (van Fraassen, 2008, p. 13, emphasis in original)
For the statue of Minerva, distortions contribute to better representation because the representation
is indexed by a perspective, i.e. the angle of view. The caricature also is indexed by a particular
perspective, though in this case not a literal viewpoint, but a stance towards Nixon as buffoonish and,
perhaps, “tricky.”
It is critical to van Fraassen’s argument that perspective is a matter of choice. Furthermore, this
choice cannot be found within the object itself—the representational properties of an object may change
with its use. Van Fraassen illustrates this point by considering a famous photograph of the Eiffel Tower
by Doisneau.
If I send it to you from Paris as a postcard, with the single note “Wish you were here!”,
then it is itself a photo of the Eiffel Tower. If I insert it into the book I am writing about
photography, then it represents the famous photo by Doisneau . . . In other words, if it is an
image of something at all then what it is an image of depends on the use, on what I use it to
represent. (van Fraassen, 2008, p. 21)
These considerations drive van Fraassen to state his “hauptsatz,” or “fundamental theorem,” of representation:
There is no representation except in the sense that some things are used, made, or taken, to
represent some things as thus or so. (p. 23)
It is this hauptsatz which ensures that there can be no natural representation; if we accept it, then
we must acknowledge that N-representation is not a species of representation. But notice that there are
two distinct steps leading to van Fraassen’s hauptsatz : first, the observation that representations are
inherently perspectival; second, the claim that the setting of perspective is a matter of choice or use.
I wish to accept the first step, but deny the second. If we consider the kinds of examples Goodman
considered in the context of artistic representation, e.g. caricatures, then van Fraassen’s second step
seems natural. It looks much less so, however, if we consider other examples of representation more
closely.
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Take, for instance, the example of the Mercator projection. This is a technique for making a two
dimensional map of a spherical surface—the map of the world hanging in your second grade classroom
was likely the result of a Mercator projection. One can make a Mercator projection very simply if one
has a transparent globe with the features to be mapped outlined on it in black. Wrap a piece of paper
around the globe along one of its great circles and shine a light from inside the globe. The shadow cast
on the wrapped piece of paper is the Mercator projection and can be traced. To make the map in your
second grade classroom, the paper was wrapped around the globe along the equator.
A Mercator projection is inherently perspectival, and it distorts the true pattern of lines on the surface
of the globe. The choice of which great circle around which to wrap the paper sets a perspective. Features
on the surface of the globe are more and more distorted in the projection as they get further from the point
of contact with the paper. At the limits, e.g. the poles on a typical Mercator map, distortion becomes
infinite. In particular, the poles, which are mere points on the surface of the globe, are stretched to lines
on the map which fill the entire top and bottom edges of the paper. The choice of perspective is related to
this distortion. In particular, we choose the equator, or some very close meridian, in order to minimally
distort those landmasses near the equator on which the preponderance of the human population of the
earth lives. So, it is indeed our interest in the world, the use to which we intend to put the map, which
determines the choice of perspective.
Notice, however, a crucial difference between the Mercator projection and the caricature. In the
caricature of Nixon, there are many choices to make: how long the nose should be, how slack the jowls,
how beady the eyes. There is no strong relationship between these choices; the length of the nose does
not in any way determine the slackness of the jowls. In the Mercator projection, there is a single choice to
make: which great circle around which to wrap the paper. Once this single choice is made, the distortions
of the globe’s surface in the map are entirely systematic. In the case of the Mercator projection, there
is one degree of freedom, one choice to be made, in the case of the caricature, there are many (arguably
infinite) degrees of freedom.
But N-representation is systematic in precisely this sense! This systematicity is a result of the requirement that an N-representation be induced by a causal process. A foot may step in mud at many different
angles. Once the foot does step, however, the relationship between it and the footprint is completely
determined by the angle and the properties of the mud. The relationship between the width of the heel
and the width of the back of the footprint and that between the width of the toes and the width of the
front of the footprint cannot be varied independently. Likewise, once the position of the sun is set, the
pattern of the tree’s shadow is uniquely determined. In these examples, there is a perspective needed
to establish N-representation, but that perspective is determined by a natural process, not some agent’s
conscious choice.
Van Fraassen’s argument begs the question against N-representation. He begins by considering representation in art; he observes the role of choices; and he derives his hauptsatz. If instead we begin by
looking at systematic examples of representation, such as the Mercator projection, the role of choice in
establishing the perspectival quality of representation seems much less essential. Once we observe that
shadows and footprints share essential features with the Mercator projection, it is natural to subsume
them all under a common notion of representation.
Of course, this is not to deny that van Fraassen’s considerations apply to some representations. We
might acknowledge that there is some sense in which the photo in the art book represents a famous
11
photograph by Doisneau, and not the Eiffel Tower. But there is also something about the photo which
makes it suitable to be used as a postcard. This something is captured by the fact that the photo Nrepresents the Eiffel Tower. N-representation can thus explain how an object is suitable for use as an
intentional representation. There is room in this world for more than one type of representation, and
N-representation can help us to better understand more elaborate examples of representation, whether
we find them in art, science, or the mind.
4
Mental Representation
Now that we’ve developed a philosophical account of similarity in nature and its role in supporting
representation, we can return to psychological practice and try to make some philosophical sense of it. In
particular, we’ll survey a number of psychologists who take mental representations to be N-representations
which guide behavior. By looking at a sequence of increasingly nuanced views, we’ll tease out the exact
role of similarity in this account of representation. After a brief digression on the consequences of this view
for the debate between symbolic and imagistic theories of representation, we’ll conclude by looking at the
problem of misrepresentation. The view developed here shifts the burden for explaining misrepresentation
onto an analysis of the goals and actions of the organism. Nevertheless, the theory of N-representation
can give some insight into the differential role which more or less informative mental representations play
in generating more or less appropriate actions toward a goal.
4.1
“Functioning” N-Representations
The psychologists surveyed here identify two marks of a mental representation: first, that it constitutes
an N-representation; second, that it be functioning.3 It is important to distinguish the use of the term
“function” here from its use in the philosophy literature (e.g. “functionalism” or “functional role semantics”). Let’s look at some representative passages in order to understand the notion of “functioning”
relevant for this theory of mental representation.
Stephen Palmer’s views are representative of those which can be found throughout the study of
perception. He identifies representation with a causally induced homomorphism, i.e. an N-representation:
. . . a representational system can be analyzed as a homomorphism: a mapping from objects
in one domain (the external world) to objects in another domain (the internal representation)
such that relations among objects in the external world are mirrored by corresponding relations
among corresponding objects in the representation. . . .
The causal factor underlying the homomorphism is important for two reasons. One is
that for the representation to be current, as a perceptual representation must be, it requires
constant updating. A causal chain from events in the external world to events in the internal
representation is an ideal way (though not the only way) to achieve this. The other is that
for the representation to be authentic, rather than accidental, there must be some linkage to
3 Structurally, this view bears some similarity to so-called “two factor” versions of the functional role theory of mental
representations (for a survey, see Perlman, 1997), though the details differ substantially from any pre-existing two factor
view of which I am aware. In particular, it would be a mistake to conflate “functional role” in the sense used here with
“conceptual role.” The closest philosophical analog to this view is in the work of Robert Cummins, discussed later in this
section.
12
the world it represents. Again, a causal connection seems to be the ideal solution. (Palmer,
1999, p. 77–8)
One reason for the popularity of this view in the study of perception is the overwhelming evidence for
the importance of causally induced structural homomorphisms in the physiology of perception. Cones
in the eye, for example, are arrayed across the retina like the photoreceptors in a camera. The spatial
pattern of excitation on the retina is projected through the lateral geniculate nucleus, onto the primary
visual cortex at the back of the brain, then forward through the later areas of visual processing in a
sequence of structure preserving maps. At each stage of this processing chain, the pattern of neural
activation constitutes an N-representation of the stimulus.
The N-representations in a perceptual system are not full-blown representations according to Palmer
unless they are exploited for reasoning about the world.
Causally driven structural similarity between a model and its object may be a necessary
condition for something to function as a representation, but it is not sufficient. A further
requirement is that there be processes that use it as a representation in the sense of taking it
as a surrogate of the world to which it corresponds. (Palmer, 1999, p. 78)
The mark of a functioning representation is not to be found in the intent or choice of an agent, but in the
actions the system performs. Palmer uses the example of a thermostat here, and the relation between its
internal bimetallic strip and ambient temperature. The position of the strip’s end N-represents changes
in ambient temperature. The crucial additional factor here is not what the thermostat was built for
(regulating temperature), but what it does, namely respond differentially to changes in the position of
the strip’s end. It uses the position of the strip’s end as a surrogate for the world in the sense that it
responds differentially to the position of the strip, not to changes in temperature directly.
N-representations again emerge as a natural account of representation in the study of action and
navigation. The abilities of bees, wasps, mice, etc. to navigate novel paths through familiar environments
is easily modeled on the assumption of a structured internal representation, standing in a causally-induced
homomorphism toward the physical environment. The fact that these paths are novel demonstrates
they result from the use of representations rather than mere habit. The fact that these paths depend
upon the familiarity of the environment demonstrates that they are causally induced by the organism’s
exploration of that environment. An example here is the model of Deutsch (1960), which posits that
familiar environments are represented by a cognitive network of nodes (standing for landmarks) with
connections signifying topological connectedness. Such a model can account for empirical findings about
the navigational behavior of wasps, bees, mice, etc. (see Gallistel, 1980, Ch. 11, for a detailed discussion).
Like Palmer, Gallistel requires more than mere N-representation, he requires that N-representations
be functioning, or “put to use”:
The exploitation of the correspondence to solve problems in the one domain using operations
belonging to the other establishes a functioning isomorphism: an isomorphism in which the
capacity of one system to represent another is put to use. (Gallistel, 1993, pp. 15–6)
Gallistel’s notion of functioning requires that an organism’s differential responses to an N-representation
be systematic:
13
Isomorphisms exist between systems, not between sets of entities. . . . The representing system
must perform operations on the entities generated by the mapping, and there must be a correspondence between the operations the representing system performs with the representatives
and the relations or processes in which the represented entities play roles—for example, the
correspondence between numerical addition and combining two weights on a pan of a balance.
(Gallistel, 1993, pp. 24)
In technical terms, Gallistel requires that the homomorphism between a mental representation and the
world preserve not only relations (static structure), but also operations (dynamic structure). Changes in
the mental representation must systematically correspond to changes in the world. For instance in the
Deutsch model, changing activation at nodes in the wasp’s internal representation should correspond to
changing positions within its environment.
One way to make sense of Gallistel’s notion of system is in terms of sets of N-representations. Consider
a set of N-representations such that they N-represent the environment via the same causal mechanism, and
small differences between the N-representations correspond to small differences in the environment. Now
suppose a second, internal causal factor (an operation) influences which N-representation is instantiated,
and that application of this internal operation corresponds to behavior by the organism which brings
about the corresponding state of the environment. As activation spreads through the wasp’s cognitive
map, it moves through its environment, changing which landmark is in view.
Mental representations N-represent the environment, and consequently they are objectively similar
to the environment. What role does this similarity between representation and environment play in
subserving the systematic use of mental representations described by Gallistel? A nuanced answer to this
question can be found in the work of Roger Shepard.
Shepard developed his view while performing experiments on spatial reasoning. He discovered, for
example, that the time it takes to recognize a shape varies linearly with the angle of rotation of the
stimulus shape from its initial presentation. Moreover, response times can be manipulated by asking
subjects to visualize rotation of the shape ahead of time (Shepard, 1975). Subjects can recognize details
of spatial structure with an enormous degree of precision, e.g. which angles of intersection between two
lines are possible views of the corner of a cube and which are not. They perceive continuous rotations
or transformations when presented with static stimuli in quick succession (as when two lights a small
distance apart flashing in sequence are perceived as a single light moving rapidly back and forth) (Shepard,
1981). Shepard developed a unified framework for modeling these diverse phenomena.
Shepard argues that systems of the sort described by Gallistel implement second-order isomorphisms:
[W]hat enables a given internal process to represent or to stand for a particular external
object is the way that that internal process functions within the organism—particularly in his
relation to that object and to other similar objects. The essential things are, in particular,
(a) that there be an orderly causal connection between the external objects and the internal
process that is to represent it, and (b) that internal representations corresponding to similar
external objects be functionally related so that a response learned to one will tend to generalize
to another (as required by second-order isomorphism). When, in addition, there is a degree
of first-order isomorphism, it is not there to enable the internal representation to refer to
its external object. It is there, rather, to provide the mechanism for the realization of the
14
necessary second-order isomorphism and, especially, to provide the structural information
needed for more differentiated responses of selective attention, analysis, transformation, and
manipulation. (Shepard, 1975, p. 92)
Let’s unpack this into our terminology. By Shepard’s lights, the most significant feature of a mental
representation is not that it constitute an N-representation of the environment in the static (“first-order”)
sense. More important is that sets of like objects generate the same, or closely related, responses. Insofar
as the set of like objects in the environment and the set of generated responses themselves constitute
legitimate structures for scientific study, the relationship between these two sets will also satisfy the
definition of objective similarity. Nevertheless, the relationship between these structures is importantly
different from the static structural similarities we have examined so far, it is dynamic and hypothetical.
The objective similarity here is between possible stimuli and possible responses, not a particular given
stimulus and the particular response it generates.
Consider again the example of the wasp. The wasp is familiar with its environment and has generated
an internal map which N-represents that environment. By Shepard’s lights, the mark of full representation
here is that like inputs (say, a landmark viewed from different angles) generate like responses (say, crawling
in the direction of its nest). It is this second-order map between inputs and behaviors which demonstrates
representation. The first-order map between cognitive structure and environment is important only
insofar as it subserves the second-order map. It is “the mechanism for the realization of the necessary
second-order isomorphism.”
Shepard (1981) resists calling the second-order map which marks representation a similarity, preferring
instead the term “complementarity.” His guiding analogy for thinking about complementarity is the
relationship between a key and a lock.4 This analogy nicely moves beyond Aristotle’s signet ring example
by adding a dynamical, functional component. Provided the dynamical structures which stand in this
second-order relationship continue to satisfy the requirement that they appear in the scientific view of
nature, they will still satisfy the definition of objective similarity. Thus a dynamic similarity obtains
between possible objects the organism might encounter and possible behaviors it might generate in
response.
Shepard speculates that first-order objective similarity may subserve second-order functioning complementarities between behavior and environment. The view that first-order similarity is actually necessary
for the successful representation of and response to nature has been defended at length by Robert Cummins (e.g. Cummins, 1996; Cummins and Poirer, 2004). Cummins takes the mark of a representation to
be the potential it offers for reasoning about the represented. In order to make sense of how an organism
comes to exploit a representational structure, its representational content must be present antecedent to
the organism’s use of it. This is what Cummins has called “unexploited content—content a representation
has, but which the system that harbors it is currently unable to exploit” (Cummins et al., 2006, p. 195).
Cummins argues that this unexploited content must be present first before the representational structure
can come to be used, or exploited, by some function. His solution for explaining unexploited content
is to identify representations with structures isomorphic to a target. Then the internal structure of the
representation represents the structure of the target in virtue of this isomorphism, independent of the
functional role the representing structure plays within the organism.
4 c.f.
Cummins, 1996, p. 102.
15
Despite many commonalities between the view of Cummins and that presented here, we diverge
on the preconditions for similarity. Cummins identifies representation with the mere existence of an
isomorphism between representation and represented, and does not restrict attention to causally induced
mappings as we have done. He argues that the Newman Problem (too many isomorphisms) need not
be solved in order for isomorphic representations to play the relevant explanatory role in understanding
behavior. It is the fit between a representation and the target of an organism’s actions which explains
how a representation successfully subserves action; if the representation fits additional, irrelevant targets
as well, this fact in no way undermines this explanatory role. Consequently, “[t]he fact that [the picture
theory of representation] makes representational contents non-unique, . . . presents no problem, provided
it does not make representational content indiscriminate” (Cummins, 1996, p. 102).
I’m afraid that I must agree with Cummins’ conclusion, while disagreeing with its consequences for his
view. To take the Newman Problem seriously is to realize that the isomorphism view of representational
content is indiscriminate without a prior constraint on permissible relations and mappings. Some mapping exists between any two structures. Here, we’ve insisted that homomorphisms be causally induced
in order to constrain the number of permissible mappings. Our account of representation does indeed
leave contents non-unique, but no longer indiscriminate. Taking causality as the relevant source of bias
has the further happy consequence of explaining why the unexploited content in a mental representation
represents, not just how. A mental representation represents the world via its similarity with the world
and because it was causally induced by the world.
A common theory of mental representation has emerged from the psychologists surveyed here. Mental
representations represent the world in virtue of their objective similarity to the world. This similarity
exists and is relevant because it is causally induced. The most important feature of this similarity is
that it be dynamic, i.e. that mental operations on an internal representation systematically correspond
to physical operations on the represented target in the world. This second-order similarity is most likely
(arguably, necessarily) subserved by a first-order similarity between static representations and static
structures in the world.
A final question concerns the locus of misrepresentation in the view developed here. Before we tackle
that problem, however, a brief digression on the relationship between the requirement of homomorphism
and the debate between symbolic and imagistic theories of representation.
4.2
Brief Digression: Symbols and Similarity
In the twentieth century, the rejection of similarity went hand in hand with the rise of symbolic theories
of representation. Likewise, philosophers known for championing the view that mental representations
represent in virtue of their homomorphic relationship to structures in the world are often known also for
embracing alternatives to the symbolic approach, such as connectionism (e.g. Paul Churchland, 2002,
2007). Furthermore, several of the psychologists discussed here (e.g. Shepard and Kosslyn) have defended
the importance of homomorphisms in the context of arguing for an imagistic theory of representation.
Nevertheless, it is important to emphasize that the requirement of objective similarity can be met even
by symbolic systems, and thus the considerations raised here cut across the symbolic vs. imagistic debate.
An easy example to see this is that of navigation in a familiar environment discussed above. I
claim, following Deutsch and Gallistel, that bees, wasps, etc. navigate a familiar environment using a
representation of that environment. This representation is an N-representation, and it is in virtue of the
16
objective similarity between this representation and the environment that the organism is able to navigate
successfully. Now, it so happens that Duetsch and Gallistel conceive of this representation as a network
of nodes, where each node stands for some landmark in the environment. But could we achieve the same
representational goal, preserving the requirement of objective similarity, with a symbolic representation?
Not only can this goal be achieved, but indeed it must be achieved if the symbolic system is to be
of any use for navigational purposes. Such a symbolic system might use arbitrary symbols a, b, c, etc.
to represent landmarks and some relational symbol R to represent the “next to” relation. Then, the
symbolic representation of the environment would consist of a list aRc, cRd, dRa, etc. of sentences in
this symbolic system. If it is faithful to the environment, this symbolic representation will stand in a
homomorphic relationship to it, and if it was learned, this homomorphism will be causally induced. So,
while the relationship between each symbol and each landmark will be arbitrary, the relationship between
the symbolic system as a whole and the environment will constitute an N-representation, and it is in
virtue of the objective similarity between the symbolic system and the environment that it can be used
to navigate successfully.5
Now, finally, we can resolve the puzzle of our first paragraph. Why is it that a pencil which I have
stipulated represents Canada cannot be used to reason about Canada? Because it is not part of a
structure. Suppose I stipulate this pen represents the United States, and this eraser represents Mexico.
Still, nothing can be learned about Canada, the U.S., or Mexico. Once I add structure, however, say by
organizing the pencil, pen, and eraser in a line such that each is to the left of the next, the structure as
a whole can be used for reasoning. If, for instance, I discover the homomorphism between this structure
and North America (which maps “to the left of” onto “to the North of”), I can use the structure to
behave in the world. For example, if I want to get to Canada from the U.S., I can use this representation
to discover that I should go North.
In order for reasoning to work here, it doesn’t matter whether the relation between primitive parts of
the structure is spatial (as when I arrange them on a desk) or symbolic (as when I write a list of sentences
in which a relation symbol stands for “to the North of”). What matters for successful reasoning is that
the representation have, or be part of, a structure, and that this structure stand in the correct relationship
to the world, namely that it N-represents the world. Then the objective similarity between representation
and world can be exploited to guide action.
4.3
Misrepresentation Demystified
Some philosophers take the possibility of misrepresentation as constitutive of the concept of representation. Yet the possibility of mis-X presumes a normative standard for evaluating X, whatever X might be.
This worry motivates the move to teleosemantics in, for example, the work of Dretske (1988, 1995). What
is the source of the potential for misrepresentation on the view developed here? If N-representations are
merely a consequence of causal relations, how can they possibly misrepresent?
Once an N-representation is exploited by an organism, then it can be used successfully or unsuccessfully. So, requirement that a mental representation constitute a functioning N-representation permits
at least one source of potential representational error, namely error as a mismatch between the goal
5 Of course, the use of a network rather than a symbolic system is crucial in the Deutsch and Gallistel model because the
actions of the organism are explained in terms of spreading activation along a gradient through the network. The example
considered here addresses merely the requirements on representation, not the mechanism for using representations to guide
action.
17
of the organism in using the representation and the actions it generates from that representation. Of
course, there are still puzzles about normativity here: what determines the organism’s goals; whence its
agency? However, these are not puzzles about the normativity of content. Misrepresentation of this kind
reduces entirely to misuse, and the onus for explaining representational error falls not on the theory of
representation, but on the theory of action.
This is the strategy pursued by Cummins for explaining representational error. He argues that theories
which cash representation out purely in terms of functional role cannot distinguish between content and
use. A virtue of the isomorphism approach is that it can cash content out in terms which do not appeal
to use (i.e. the existence of a mapping between representation and represented). Then the functional role
of a representation is left free to determine whether the representation is used effectively or not. When
not, then the representation is misused (Cummins, 1996, Ch. 4).
I agree with Cummins on this point. I do not take the possibility of misrepresentation as constitutive of representation. Rather, the necessary feature for a structure to constitute a representation
is that it present again information about its target. Thus, even though they cannot misrepresent, Nrepresentations are indeed representations. Furthermore, once an N-representation is used, it may be
misused, and therefore functioning N-representations in the sense discussed above may indeed “misrepresent” in the sense of being used to generate actions in a manner which is not efficacious for the organism’s
ends.
Consider a thermostat with faulty wiring such that it blows warm air if the ambient temperature is
warmer than its preset target temperature and cold air if ambient temperature is cooler. The bimetallic
strip in this thermostat N-represents the environment just as effectively as in a properly functioning
thermostat. Furthermore, the thermostat responds to that representation systematically, i.e. it is indeed
a functioning N-representation. Nevertheless, once we introduce an analysis of the goal of the thermostat,
we can see that it is misusing this N-representation. This diagnosis depends on our having different
analyses of the representational content of changes in the bimetallic strip and of the functional role
these changes play in determining the operation of the thermostat. Whether we call the position of the
bimetallic strip a representation or not, it contains information about ambient temperature which may
be exploited, i.e. its possible positions N-represent possible ambient temperatures.
Nevertheless, there is more to be said about the relationship between N-representations and error.
Although N-representations may not misrepresent, they may represent more or less information about
the target. The directional, perspectival character of N-representations discussed in Section 3.3 ensures
that representations of the same type, i.e. induced by the same causal process, may contain very different
quantities of information about the target. In some cases, this feature alone is enough to explain an
instance of apparent misrepresentation.
Consider the example in Figure 1.6 If we accept the definition of objective similarity given above, then
the same clause which ensures that footprints made by the same foot are similar implies that these two
shadows are similar. The radical difference in perspective set by the two sources of illumination, however,
has produced two quite different shadows of the cylinder: a square and a ring. These two shadows are
not even topologically equivalent—the objective similarity which exists between them is very coarse (i.e.
the relevant homomorphism preserves very little structure).
6 This example was inspired by the cover of Douglas Hofstadter’s Gödel, Escher, Bach, which features wooden blocks
which cast the shadow of a “G” when illuminated along one axis, an “E” when illuminated on another, and a “B” when
illuminated along the third.
18
Figure 1: Perspective can produce misleading instances of objective similarity.
The shape of the cylinder constrains the possible shadows it can cast. In this regard, the square is
very special as a cylinder shadow. The cylinder will cast a square shadow only if a light source is oriented
along the horizontal plane which bisects the cylinder widthwise. As soon as the angle or position of the
light source moves from that plane, the curves of the cylinder begin to enter into its shadow. A very
small subset of possible perspectives will generate square shadows from a cylinder.
The crucial point here is that the perspectival nature of N-representation ensures variance in the
possible N-representations of an object induced by a single causal process. Along with variance in the
nature of these N-representations comes also variance in which features are informative and which are
not. I may take a smooth footprint as indicative that the boot which made it has no tread, but it also may
be the case that the low viscosity of the mud erased the treadmarks as soon as the boot was lifted. Is it
appropriate to say the smooth footprint mis-N-represents the boot? No, but it does misleadingly represent
the boot in the sense that surface properties of the base of a footprint can be information-bearing, but
in this case they are not.
The natural variance amongst N-representations suggests the possibility that the appearance of error
may emerge in purely mechanical responses to veridical information. Consider, for example, a machine
with a simple perceptual system, which responds only to light and dark. It inhabits a world of cylinders
and cubes, and mechanically responds to a shadow cast on its surface by generating an internal representation of a cube or a cylinder. Square shadows may be cast by either a cube or a cylinder, but cubes cast
square shadows when illuminated from more perspectives than cylinders. Thus, if our machine generates
cube-representations internally whenever a square shadow is cast upon it, it will generate a cube whenever its perceptual system is in the position of the right wall in Figure 1. If it is structured, this internal
cube-representation will constitute an N-representation of the cylinder. From an outside perspective, this
appears to be an instance of misrepresentation, even though no norms regulate the mechanical system.
Strictly speaking, all that has occurred is that some features of the internal representation which are
often information bearing (e.g. its topological properties) in this case fail to bear information about the
target.
Many optical illusions may be explained in just this way. The Ames room, for example, manipulates
19
the visual system into perceiving two men who are the same height as very different in height. It does
this by placing one man much further away than the other, but in a specially constructed environment
that appears when viewed from a specific vantage point like a regular room, with the men equal distances
from the viewer. The visual system is “fooled” by the unusual structure of the special room into assigning
different heights to the two men. But it is not as if any normativity is required to generate this “error”—it
is a simple consequence of the hardwiring of the visual system. Features of the internal representation
which in most cases are information bearing, e.g. distance between the top of a man’s head and the ceiling,
in this unusual case fail to bear information. What appears from the outside as a misrepresentation (of
height) can be explained internally in terms of mere mechanical N-representation.
5
Conclusion
The goal of this paper has been to reconcile psychological practice with philosophical theory. In particular,
the psychological view that mental representations can be used to reason about objects in the world
because they are similar to those objects contradicts the philosophical view that similarity is irrelevant
for representation. By assuming causal structure in the world, we solved two problems in sequence.
First, we avoided the problem of too many similarities, defining objective similarity in terms of causally
induced homomorphisms. Second, by restricting attention to only those objective similarity relations
directed along the flow of causal influence, we defined a coherent notion of representation in nature, or
N-representation.
In surveying psychological theories of perception, navigation, and spatial reasoning, we saw mental
representations identified with functioning N-representations. On this account, the objective similarity
between a mental representation and its target explains its utility for guiding action; the fact that
the representation was caused by its target explains its relevance for guiding action; and the use of
the representation for guiding action locates a source for potential error in the potential for misuse.
Furthermore, we demonstrated that these considerations are orthogonal to the debate between symbolic
and imagistic accounts of representation.
I hope the account of objective similarity developed here has made similarity safe again for the study
of mental representation, and perhaps even for the study of representation in general. At the very least,
our discussion has taken assumptions about representation implicit in the psychological literature and
made them explicit, clearing the way for further debate.
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